Wiring – Why does adding a C wire for a thermostat blow the fuse

You moved the Green to C on one end but the picture with the transformer in the top left shows Green still on G there it should be on C.

The picture below you would pull the green wire off of Green at both the stat and furnace and move it to C/Common, then you jumper Yellow to Green at the furnace.

There should be no problem connecting the C wire from your thermostat to the C terminal in the furnace. Take a look at this diagram, which is a rough approximation of your system.

Notice the cable going to the condensing unit has a red wire connected to the Y terminal in the furnace, and a white wire attached to the C terminal. When the thermostat calls for COOL, it does so by energizing the Y terminal (connecting R to Y). This provides power to the coil of the contactor, which allows it to close the contacts and turn the condensing unit on. The other side of the contactor coil is connected to the C terminal, as to provide a complete circuit.

You'll notice in the first image that two relays in the furnace, the thermostat, and the contactor coil all connect to the same leg of the transformer. This is because the thermostat basically works like a switch, connecting the R wire to the C wire through control circuitry. Anything that needs 24VAC power, will be connected to both legs of the transformer.